A regularization theorem for bounded-degree self-maps

Abstract

Let $K$ be an algebraically closed field of arbitrary characteristic and let $X$ be an irreducible projective variety over $K$. Let $G\subseteq\text{Bir}(X)$ be a bounded-degree subgroup. We prove that there exists an irreducible projective variety $Y$ birational to $X$, such that every element of $G$ becomes an automorphism of $Y$ after the birational transformation. If $K=\mathbb{C}$, this result is stated in [Can14, Theorem 2.5] and the proof backs to [HZ96, Section 5]. The proof in [HZ96] is not purely algebraic. Inheriting the methods in [HZ96], we give a purely algebraic proof of this statement in arbitrary characteristic. We will also discuss a corollary of this result which is useful in arithmetic dynamics.